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2009 AMC 8 Problems/Problem 20: Difference between revisions

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==Problem==
==Problem==
How many non-congruent triangles have vertices at three of the eight points in the array shown below?
How many non-congruent triangles have vertices at three of the eight points in the array shown below?


<asy>dot((0,0));
<asy>dot((0,0));
dot((0,.5));
dot((.5,.5));
dot((.5,0));
dot((.5,0));
dot((.5,.5));
dot((.0,.5));
dot((1,0));
dot((1,0));
dot((1,.5));
dot((1,.5));
Line 15: Line 14:


==Solution==
==Solution==
Assume the base of the triangle is on the bottom four points, because a congruent triangle can be made by reflecting the base on the top four points. For a triangle with a base of length <math>1</math>, there are <math>3</math> triangles. For a triangle with a base of length <math>2</math>, there are <math>3</math> triangles. For length <math>3</math>, there are <math>2</math>. In total, the number of non-congruent triangles is <math>3+3+2=\boxed{\textbf{(D)}\ 8}</math>.
Assume the base of the triangle is on the bottom four points because a congruent triangle can be made by reflecting the base on the top four points. For a triangle with a base of length <math>1</math>, there are <math>3</math> triangles. For a triangle with a base of length <math>2</math>, there are <math>3</math> triangles. For length <math>3</math>, there are <math>2</math>. In total, the number of non-congruent triangles is <math>3+3+2=\boxed{\textbf{(D)}\ 8}</math>.


==See Also==
==See Also==
{{AMC8 box|year=2009|num-b=19|num-a=21}}
{{AMC8 box|year=2009|num-b=19|num-a=21}}
{{MAA Notice}}
{{MAA Notice}}

Latest revision as of 18:19, 16 January 2025

Problem

How many non-congruent triangles have vertices at three of the eight points in the array shown below?

[asy]dot((0,0)); dot((.5,.5)); dot((.5,0)); dot((.0,.5)); dot((1,0)); dot((1,.5)); dot((1.5,0)); dot((1.5,.5));[/asy]

$\textbf{(A)}\ 5 \qquad \textbf{(B)}\ 6 \qquad \textbf{(C)}\ 7 \qquad \textbf{(D)}\ 8 \qquad \textbf{(E)}\ 9$

Solution

Assume the base of the triangle is on the bottom four points because a congruent triangle can be made by reflecting the base on the top four points. For a triangle with a base of length $1$, there are $3$ triangles. For a triangle with a base of length $2$, there are $3$ triangles. For length $3$, there are $2$. In total, the number of non-congruent triangles is $3+3+2=\boxed{\textbf{(D)}\ 8}$.

See Also

2009 AMC 8 (Problems • Answer Key • Resources)
Preceded by
Problem 19 		Followed by
Problem 21
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25
All AJHSME/AMC 8 Problems and Solutions

These problems are copyrighted © by the Mathematical Association of America, as part of the American Mathematics Competitions. Error creating thumbnail: Unable to save thumbnail to destination

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